Refractors:

The earliest method was to bend the rays by passing them through one or more pieces of glass which had curved, polished surfaces, this is called a Refractor.

Advantages of Refractors:

Enclosed tube: dust and moisture doesn’t enter the optical tube.

Fixed optics: don’t require routine collimation  (optics don’t have to be adjusted by the user)

No central obstruction: Less loss of light entering the tube, no alteration of the diffraction pattern

Produce high-contrast, fine-resolution images ideal for planetary viewing

Disadvantages of Refractors:

Many wavelengths of light are passing through glass, uneven bending of the rays causes false color, around bright objects
 
False color is eliminated with additional lenses and special glass. Requires at least four very accurately shaped, polished and coated lens surfaces
 
More expensive to produce than other telescope designs

Inexpensive refractors tend to add false color to images

Reflectors:

The second method of focusing light is to reflect the rays off of the surface of a curved mirror, producing a type of telescope called a reflector
 
Most common reflectors in use today are called Newtonians, a design pioneered by Isaac Newton

Rather than lenses, the reflector uses 2 mirrors to bring light to the eyepiece

Advantages of Reflectors:
 
Dollar for dollar, reflectors offer the most aperture. They produce sharp images that are free of any added color

Disadvantages of Reflectors:

The spider holding the secondary mirror forms a central obstruction that produces a diffraction pattern

Due to the shape of the primary mirror, reflectors suffer from a condition called coma. Objects at the outside edge of the field of view have the appearance of being wedge shaped or look like little comets

Some eyepiece designs work well to counteract this effect

Catadioptric Telescopes: